Find a basis and the dimension of the solution space of the homogeneous system

$2x_{1} + x_{2} \ \ \ \ \ \ \ \ = 0 \\$ $x_{1} + x_{2} - x_{3} = 0 \\$ $\ \ \ \ -x_{2} + 2x_{3} = 0$

Question

Find a basis and the dimension of the solution space of the homogeneous system

[latex]2x_{1} + x_{2} \ \ \ \ \ \ \ \ = 0 \[/latex] [latex]x_{1} + x_{2} - x_{3} = 0 \[/latex] [latex]\ \ \ \ -x_{2} + 2x_{3} = 0[/latex]

Accepted Answer

We found in Example 2.2.9 that the general solution of this system is

[latex] \vec{x} = t\left [ \begin{matrix} -1 \ 2 \ 1 \end{matrix} ight ] , \ \ \ \ t \in \mathbb{R}[/latex]

This shows that a spanning set for the solution space is [latex]\mathfrak{B} = \left\{\left [ \begin{matrix} -1 \ 2 \ 1 \end{matrix} ight ] ight\}[/latex]. Since B contains one non-zero vector, it is also linearly independent and hence a basis for the solution space. Since the basis contains 1 vector, we have that the dimension of the solution space is 1.

Question 2.3.9: Find a basis and the dimension of the solution space of the ...

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